Refined $\mathrm{SU}(3)$ Vafa–Witten invariants and modularity
2018; Volume: 14; Issue: 3-4 Linguagem: Inglês
10.4310/pamq.2018.v14.n3.a3
ISSN1558-8602
AutoresLothar Göttsche, Martijn Kool,
Tópico(s)Advanced Mathematical Identities
ResumoWe conjecture a formula for the refined SU(3) Vafa-Witten invariants of any smooth surface S satisfying H 1 (S, Z) = 0 and p g (S) > 0. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions.We prove that our formula satisfies a refined S-duality modularity transformation.We provide evidence for our formula by calculating virtual χ ygenera of moduli spaces of rank 3 stable sheaves on S in examples using Mochizuki's formula.Further evidence is based on the recent definition of refined SU(r) Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank 3).
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